Is there a way to use 'render region' with numeric input for size & position?
You can turn on the Python Console Shift+F4 and define the border there.

Use commands:

bpy.data.scenes['Scene'].render.border_min_x
bpy.data.scenes['Scene'].render.border_max_x
bpy.data.scenes['Scene'].render.border_min_y
bpy.data.scenes['Scene'].render.border_max_y


Like so:



You can also turn on the Amaranth Add-on that comes with Blender (Have the 'Community' button ON) in order to see how large the Border render box is: